Journal of Signal and Information Processing, 2011, 2, 266-269
doi:10.4236/jsip.2011.24037 Published Online November 2011 (http://www.SciRP.org/journal/jsip)
1
Fast Algorithm for DOA Estimation with Partial
Covariance Matrix and without
Eigendecomposition
Jianfeng Chen1, Yuntao Wu2*, Hui Cao2, Hai Wang2
1The 54th Institute of China Electronics Technology Group Corporation, Shijiazhuang, China; 2Key Laboratory of Intelligent Robot
in Hubei Province, Wuhan Institute of Technology, Wuhan, China.
Received July 17th, 2011; revised September 10th, 2011; accepted September 18th, 2011.
ABSTRACT
A fast algorithm for DOA estim ation without eigend ecomposition is proposed. Unlike th e availab le propagation method
(PM), the proposed method need only use partial cross-correlation of array output data, and hence the computational
complexity is further reduced. Moreover, the proposed method is suitable for the case of spatially nonuniform colored
noise. Simulation results show the performance of the proposed method is comparable to those of the existing PM
method and the standa rd MUSIC method.
Keywords: Fast Algorithm, DOA Estimation, Subspace-Based Method
1. Introduction
DOA estimation of spatial signal source with an array of
sensors has been an active research problem in array sig-
nal processing due to its wide applications in radar, sonar
and so on. Many classical algorithms have been devel-
oped in the past thirties years [1-3], in particular, a class
of subspace-based methods such as MUSIC [1], Root-
MUSIC [2], and ESPRIT [3] are drawn more attractive
due to its higher resolution performance but without mul-
tiple-dimension search computation. However, most of
the subspace-based methods are required to compute the
eigendecomposition of covariance matrix of array output
data in order to obtain the so-called signal subspace or
noise subspace, which its application is limited in case of
larger number of array sensors. To avoid the computa-
tional load of the eigendecomposition of covariance ma-
trix, in recent years, some fast algorithms for DOA esti-
mation have been proposed for certain condition in the
literature [4-7]. In particular, the propagation method (PM)
[6,7] without eigendecomposition has been discussed due
to lower computational load. However, the available PM
method need use the whole covariance of array output
data to obtain the propagation operator, therefore, the
PM-based algorithm is only suitable to the presence of
white Gaussian noise, and its performance will be de-
graded in spatial nonuniform colored noise [8].
In this paper, we present a modified PM algorithm for
DOA estimation with an ULA, a different computation
method for the propagation operator is given, which is
only obtained by the partially cross-correlation of array
output data. As a result, the proposed algorithm is com-
putationally simpler than the available PM method [6].
Moreover, the proposed algorithm is suitable for the case
of spatially nonuniform colored noise due to using the
off-diagonal elements of array covariance matrix.
2. Proposed Method
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Let an uniform linear array of N sensors receive P
narrow band signals impinging from the sources with
unknown spatial DOA’s


1,,
P

 . The sensor array
outputs can be expressed as:






,1,2,,tttt  xAsn L (1)

Fast Algorithm for DOA Estimation with Partial Covariance Matrix and without Eigendecomposition267

where and
denote the received
array data vector and the array manifold matrix, respec-
tively. and
stand for the source wave-
form vector and sensor noise vector.
 
1,, T N
txt xt

x
 
12
,,,
P
   

Aaa a
 
1,, T P
tst st

s
 
1,, T N
tnt nt



t

n
 
 
1,exp2 πdsin, ,
exp2π1dsin
ii
T
i
j
jN






a
is the steering vector and denotes the transposition
of a matrix. The sensors noise is assumed to be a zero
mean spatially and temporally white Gaussian process.

T

The following assumptions are made in the subsequent
developments:
(A1) The number of sources P is known a priori and the
number of sensors satisfies N > 2P.
(A2) The set of P steering vectors is linearly independent
and the P signal sources are statistically independent of
each other.
Under the assumption of N > 2P, the array manifold
matrix A can be partitioned as follows:
123
,, T
TTT

 AAAA


(2)
where i, i=1,2,3 is a matrix with dimension P × P, P ×
P, (N – 2P) × P, respectively.
A
Based on the above partition of A and using Equation
(1), the following partially cross-correlation matrices of
the array output are defined as:
 

 
121 2
1:,:1 :2,:
H
H
ss
Et PtPP



Rx xAR

A
(3)
 

 
313 1
21:,:1:,:
H
H
ss
Et PNtP

 

Rxx AR


A
(4)
  

  

32
32
21:,: 1:2,:
H
H
ss
Et PNtPP

 


Rx x
AR A


(5)
where takes the i-th to j-th row of
 
:,:tijx


tx,

H

denotes the Hermitian transpose and




ts
H
ss Et


Rs 

is the source signal covariance matrix.
Under the assumption of (A2), both
s
s and i,
i = 1,2 are invertible matrices, therefore, the following
equation holds:
R A

1
11
32121 32211 3
HH
ss ss


RRA ARAARAA A
1


(6)
Similarly, the following equation can also be obtained:
1
31 21 23
RRAA
(7)
Combines Equation (6) and Equation (7) and yields:
11
3212 13121 23
2

RRA RRAA (8)
Equivalently, the above equation can be written as:
11
32 1231 212
2NP





RRRRIA0 (9)
where 2
N
P
I
2
is an identical matrix with dimension N –
2P and
N
P
0
H
a zero matrix, respectively.
Let 11
32 1231 212
2NP







QRRRRI and we have
H

QA 0 (10)
which implies that the columns of
H
Q form a basis of
the null space of A, i.e.,

 
1, 2,,
HkkP
 Qa 0
Using the estimated matrix from the finite array
output data in real application, similar to the MUSIC-
based method, we may form the following spatial spec-
trum function and obtain the estimates of
ˆ
Q
(1,2,,
kk}P


 from P spectrum peaks:
 
2
1
ˆH
f 

Qa (11)
where




 
1,exp2πdsin, ,
exp2π1dsin
ii
T
i
j
jN






a
Alternately, using the Root-MUSIC-based method [2]
to get the estimation value of k

directly.
It is worthy to note that the estimation of need not
any eigendecomposition, and the noise information is not
involved in , therefore, the proposed method can be
used in the case of spatial non-uniform noise or spatial
band limited noise [8].
ˆ
Q
ˆ
Q
Regarding major computational complexity, the num-
ber of multiplications for calculating includes P(N -
P)L in the cross-correlation computation in Equations
(3)-(4) and O(P3) for the inversion of 12 in Equations
(6)-(7), respectively, while the MUSIC method [1] in-
volves N2L for covariance matrix computation and O(N3)
for the eigenvalue decomposition of the resultant covari-
ance matrix. On the other hand, the number of multipli-
cations for computing the propagation operator in
the available PM method [6] is NPL + O(P3). Apparently,
the computational complexity of the proposed algorithm
is lower than those of the MUSIC method and the pro-
pagation method.
ˆ
Q
R
Q
3. Simulation Results
In the first simulation, the experiments are performed
with an uniform linear array (ULA) with N = 10 sensors

Fast Algorithm for DOA Estimation with Partial Covariance Matrix and without Eigendecomposition 268
and half-wavelength inter-element spacing. Two equally
powered narrow-band sources with DOA’s 17



and 2 impinge on the array and the two sources
are statistically independent of each other. Let L = 100
and the average results for 200 independent runs are used
to evaluate the estimation performance of different meth-
ods.
8  
Figures 1-2 show the RMSEs (root-mean-square error)
for DOA estimation versus different SNR conditions.
The results using the conventional MUSIC algorithm [1],
the Standard ESPRIT method [3] and the available or-
thonormalisation PM (OPM) [7] method are also in-
cluded to contrast the performance of the proposed algo-
rithm. It is seen that the estimation accuracy of the pro-
posed method is comparable to those of the subspace-
based MUSIC method, ESPRIT and OPM method at all
the SNRs.
In the second experiment, we assume




12
,5,  6 and other parameters is same as the
above experiment, however, the spatially nonuniform
independent sensor noise has the following covariance
matrix:


2diag1,1.2, 4,12,11, 3, 0.4,10, 2
n
 Q.
The definition of signal-noise-ratio (SNR) is the same
as that in [8]. The estimation results of the proposed
method for 10 independent runs in the case of SNR =
5dB are ploted in Figure 3. It is seen that the proposed
method can resolve accurately two closely spatial sources
in the presence of spatially nonuniform noise.
4. Conclusions
A computationally efficient algorithm for DOA estimation
with an uniform linear array has been presented. The par-
tial cross-correlation of array outputs is utilized to compute
the propagation operator, and hence the proposed method
is suitable to the case of spatially non-uniform noise. Fi-
nally, it is shown that the estimation performance of the
Figure 1. RMSEs of θ1 versus SNR.
Figure 2. RMSEs of θ2 versus SNR.
Figure 3. Spectrum of proposed method in spatially non-
uniform colored noise.
proposed algorithm is comparable to those of the available
PM method as well the conventional MUSIC method at
sufficiently higher SNR conditions.
5. Acknowledgements
The work described in this paper was jointly supported
by a grant from the National Natural Science Foundation
of China (Project No. 60802046) and the Research Plan
Project of Hubei Provincial Department of Education
(No. Q20091501).
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